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Theorem nb3grprlem1 26855
 Description: Lemma 1 for nb3grpr 26857. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Hypotheses
Ref Expression
nb3grpr.v 𝑉 = (Vtx‘𝐺)
nb3grpr.e 𝐸 = (Edg‘𝐺)
nb3grpr.g (𝜑𝐺 ∈ USGraph)
nb3grpr.t (𝜑𝑉 = {𝐴, 𝐵, 𝐶})
nb3grpr.s (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
Assertion
Ref Expression
nb3grprlem1 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))

Proof of Theorem nb3grprlem1
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nb3grpr.s . . . . . . 7 (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))
2 prid1g 4564 . . . . . . . 8 (𝐵𝑌𝐵 ∈ {𝐵, 𝐶})
323ad2ant2 1114 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐵 ∈ {𝐵, 𝐶})
41, 3syl 17 . . . . . 6 (𝜑𝐵 ∈ {𝐵, 𝐶})
54adantr 473 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ {𝐵, 𝐶})
6 eleq2 2848 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
76eqcoms 2780 . . . . . 6 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
87adantl 474 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
95, 8mpbid 224 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ (𝐺 NeighbVtx 𝐴))
10 nb3grpr.g . . . . . 6 (𝜑𝐺 ∈ USGraph)
11 nb3grpr.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
1211nbusgreledg 26828 . . . . . . 7 (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐵, 𝐴} ∈ 𝐸))
13 prcom 4536 . . . . . . . . 9 {𝐵, 𝐴} = {𝐴, 𝐵}
1413a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐵, 𝐴} = {𝐴, 𝐵})
1514eleq1d 2844 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐵, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
1612, 15bitrd 271 . . . . . 6 (𝐺 ∈ USGraph → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
1710, 16syl 17 . . . . 5 (𝜑 → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
1817adantr 473 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐵} ∈ 𝐸))
199, 18mpbid 224 . . 3 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ 𝐸)
20 prid2g 4565 . . . . . . . 8 (𝐶𝑍𝐶 ∈ {𝐵, 𝐶})
21203ad2ant3 1115 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 ∈ {𝐵, 𝐶})
221, 21syl 17 . . . . . 6 (𝜑𝐶 ∈ {𝐵, 𝐶})
2322adantr 473 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ {𝐵, 𝐶})
24 eleq2 2848 . . . . . . 7 ({𝐵, 𝐶} = (𝐺 NeighbVtx 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2524eqcoms 2780 . . . . . 6 ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2625adantl 474 . . . . 5 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
2723, 26mpbid 224 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
2811nbusgreledg 26828 . . . . . . 7 (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
29 prcom 4536 . . . . . . . . 9 {𝐶, 𝐴} = {𝐴, 𝐶}
3029a1i 11 . . . . . . . 8 (𝐺 ∈ USGraph → {𝐶, 𝐴} = {𝐴, 𝐶})
3130eleq1d 2844 . . . . . . 7 (𝐺 ∈ USGraph → ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
3228, 31bitrd 271 . . . . . 6 (𝐺 ∈ USGraph → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3310, 32syl 17 . . . . 5 (𝜑 → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3433adantr 473 . . . 4 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐴, 𝐶} ∈ 𝐸))
3527, 34mpbid 224 . . 3 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐶} ∈ 𝐸)
3619, 35jca 504 . 2 ((𝜑 ∧ (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))
37 nb3grpr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
3837, 11nbusgr 26824 . . . . 5 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
3910, 38syl 17 . . . 4 (𝜑 → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
4039adantr 473 . . 3 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸})
41 nb3grpr.t . . . . . . . . . 10 (𝜑𝑉 = {𝐴, 𝐵, 𝐶})
42 eleq2 2848 . . . . . . . . . 10 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
4341, 42syl 17 . . . . . . . . 9 (𝜑 → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
4443adantr 473 . . . . . . . 8 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉𝑣 ∈ {𝐴, 𝐵, 𝐶}))
45 vex 3412 . . . . . . . . . . 11 𝑣 ∈ V
4645eltp 4494 . . . . . . . . . 10 (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶))
4711usgredgne 26681 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → 𝐴𝑣)
48 df-ne 2962 . . . . . . . . . . . . . . . . 17 (𝐴𝑣 ↔ ¬ 𝐴 = 𝑣)
49 pm2.24 122 . . . . . . . . . . . . . . . . . . 19 (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5049eqcoms 2780 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵𝑣 = 𝐶)))
5150com12 32 . . . . . . . . . . . . . . . . 17 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5248, 51sylbi 209 . . . . . . . . . . . . . . . 16 (𝐴𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5347, 52syl 17 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶)))
5453ex 405 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5510, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5655adantr 473 . . . . . . . . . . . 12 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵𝑣 = 𝐶))))
5756com3r 87 . . . . . . . . . . 11 (𝑣 = 𝐴 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
58 orc 853 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑣 = 𝐵𝑣 = 𝐶))
59582a1d 26 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
60 olc 854 . . . . . . . . . . . 12 (𝑣 = 𝐶 → (𝑣 = 𝐵𝑣 = 𝐶))
61602a1d 26 . . . . . . . . . . 11 (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6257, 59, 613jaoi 1407 . . . . . . . . . 10 ((𝑣 = 𝐴𝑣 = 𝐵𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6346, 62sylbi 209 . . . . . . . . 9 (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6463com12 32 . . . . . . . 8 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6544, 64sylbid 232 . . . . . . 7 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 → ({𝐴, 𝑣} ∈ 𝐸 → (𝑣 = 𝐵𝑣 = 𝐶))))
6665impd 402 . . . . . 6 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) → (𝑣 = 𝐵𝑣 = 𝐶)))
67 eqid 2772 . . . . . . . . . . . . . . . . . 18 𝐵 = 𝐵
68673mix2i 1314 . . . . . . . . . . . . . . . . 17 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
691simp2d 1123 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑌)
70 eltpg 4491 . . . . . . . . . . . . . . . . . 18 (𝐵𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
7169, 70syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)))
7268, 71mpbiri 250 . . . . . . . . . . . . . . . 16 (𝜑𝐵 ∈ {𝐴, 𝐵, 𝐶})
7372adantr 473 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
74 eleq1 2847 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
7574bicomd 215 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐵 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7675adantl 474 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐵) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
7773, 76mpbid 224 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
7842bicomd 215 . . . . . . . . . . . . . . . 16 (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
7941, 78syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
8079adantr 473 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
8177, 80mpbid 224 . . . . . . . . . . . . 13 ((𝜑𝑣 = 𝐵) → 𝑣𝑉)
8281ex 405 . . . . . . . . . . . 12 (𝜑 → (𝑣 = 𝐵𝑣𝑉))
8382adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵𝑣𝑉))
8483impcom 399 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣𝑉)
85 preq2 4538 . . . . . . . . . . . . . . 15 (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
8685eleq1d 2844 . . . . . . . . . . . . . 14 (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8786eqcoms 2780 . . . . . . . . . . . . 13 (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
8887biimpcd 241 . . . . . . . . . . . 12 ({𝐴, 𝐵} ∈ 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
8988ad2antrl 715 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ 𝐸))
9089impcom 399 . . . . . . . . . 10 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
9184, 90jca 504 . . . . . . . . 9 ((𝑣 = 𝐵 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
9291ex 405 . . . . . . . 8 (𝑣 = 𝐵 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
93 tpid3g 4576 . . . . . . . . . . . . . . . . . 18 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
94933ad2ant3 1115 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
951, 94syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐶 ∈ {𝐴, 𝐵, 𝐶})
9695adantr 473 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐶) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
97 eleq1 2847 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
9897bicomd 215 . . . . . . . . . . . . . . . 16 (𝑣 = 𝐶 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
9998adantl 474 . . . . . . . . . . . . . . 15 ((𝜑𝑣 = 𝐶) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
10096, 99mpbid 224 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
10179adantr 473 . . . . . . . . . . . . . 14 ((𝜑𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣𝑉))
102100, 101mpbid 224 . . . . . . . . . . . . 13 ((𝜑𝑣 = 𝐶) → 𝑣𝑉)
103102ex 405 . . . . . . . . . . . 12 (𝜑 → (𝑣 = 𝐶𝑣𝑉))
104103adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶𝑣𝑉))
105104impcom 399 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → 𝑣𝑉)
106 preq2 4538 . . . . . . . . . . . . . . 15 (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
107106eleq1d 2844 . . . . . . . . . . . . . 14 (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
108107eqcoms 2780 . . . . . . . . . . . . 13 (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝑣} ∈ 𝐸))
109108biimpcd 241 . . . . . . . . . . . 12 ({𝐴, 𝐶} ∈ 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
110109ad2antll 716 . . . . . . . . . . 11 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ 𝐸))
111110impcom 399 . . . . . . . . . 10 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → {𝐴, 𝑣} ∈ 𝐸)
112105, 111jca 504 . . . . . . . . 9 ((𝑣 = 𝐶 ∧ (𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸))) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸))
113112ex 405 . . . . . . . 8 (𝑣 = 𝐶 → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
11492, 113jaoi 843 . . . . . . 7 ((𝑣 = 𝐵𝑣 = 𝐶) → ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
115114com12 32 . . . . . 6 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣 = 𝐵𝑣 = 𝐶) → (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)))
11666, 115impbid 204 . . . . 5 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → ((𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸) ↔ (𝑣 = 𝐵𝑣 = 𝐶)))
117116abbidv 2837 . . . 4 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)})
118 df-rab 3091 . . . 4 {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝑣 ∣ (𝑣𝑉 ∧ {𝐴, 𝑣} ∈ 𝐸)}
119 dfpr2 4454 . . . 4 {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵𝑣 = 𝐶)}
120117, 118, 1193eqtr4g 2833 . . 3 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → {𝑣𝑉 ∣ {𝐴, 𝑣} ∈ 𝐸} = {𝐵, 𝐶})
12140, 120eqtrd 2808 . 2 ((𝜑 ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)) → (𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶})
12236, 121impbida 788 1 (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∨ wo 833   ∨ w3o 1067   ∧ w3a 1068   = wceq 1507   ∈ wcel 2048  {cab 2753   ≠ wne 2961  {crab 3086  {cpr 4437  {ctp 4439  ‘cfv 6182  (class class class)co 6970  Vtxcvtx 26474  Edgcedg 26525  USGraphcusgr 26627   NeighbVtx cnbgr 26807 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-2o 7898  df-oadd 7901  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-dju 9116  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-n0 11701  df-xnn0 11773  df-z 11787  df-uz 12052  df-fz 12702  df-hash 13499  df-edg 26526  df-upgr 26560  df-umgr 26561  df-usgr 26629  df-nbgr 26808 This theorem is referenced by:  nb3grpr  26857  nb3grpr2  26858  nb3gr2nb  26859
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